We now consider two ways of associating a generalized metric space to a
precubical set, and study the properties of the resulting space. Since here we
are mainly concerned about precubical sets arising as the cubical semantics of
concurrent programs, we study mainly finite geometric precubical sets, but try
to provide more general hypothesis when these generalizations are easy to
make. Part of the current section is a reformulation – and generalization – in
the categorical language of properties studied in the framework of polyhedral
complexes [1]}. The two main results of this section are
that metric and topological geometric realization provide comparable
constructions (Theorem REF ) and that realization of
precubical sets are CAT(0) cubical complexes (Theorem REF ).
